Lesson video

Hi there, my name is Ms. Lambell.

You've made a really good choice deciding to join me today to do some maths.

Come on, let's get started.

Welcome to today's lesson.

The title of today's lesson is Multiplicative Relationships and Direct Proportion, and that's within the unit, Understanding Multiplicative Relationships, Percentages and Proportion.

By the end of this lesson, you'll be able to appreciate the connection between multiplicative relationships and direct proportion.

Let's get going.

Something to remind ourselves of, because we're gonna be talking about direct proportion today, our two variables are in direct proportion if they have a constant multiplicative relationship.

Today's lesson, I'm gonna split into two learning cycles and the first one we're gonna concentrate on what we mean by direct proportion, and in the second one we're going to have a look at what the graphs of rep direct proportion look like.

We'll get going on the first one, direct proportion.

Now we are really familiar with multiplicative relationships in a variety of situations.

So for example, ones that you will probably have come across are converting between units of measurement, so converting between centimetres and metres, or litres and millilitres, or kilogrammes and grammes.

Recipes, when we scale recipes, so if a recipe is for 10 cookies but we need to make 20 cookies, we can scale that recipe using a multiplicative relationship.

Converting between units of currency using an exchange rate.

We know that there is a multiplicative relationship between, for example, the number of pounds you get for the number of dollars, and vice versa, or the number of euros you'll get for pounds.

Buying different amounts of an item.

So as long as we are buying the same item, then the number of items we buy has a multiplicative relationship to the cost of those items. If two variables have a constant multiplicative relationship, they are in direct proportion.

Like I said, we're really familiar with multiplicative relationships, so all we are really now doing is recognising that this is showing a directly proportional relationship, they are in direct proportion.

Notice I've highlighted the word there, "constant", so it needs to be a constant multiplicative relationship.

We know that any variables which have a multiplicative relationship can be represented in a variety of ways.

Can you remember the different ways of representing multiplicative relationships? I'm going to pause a moment and let you have a think and see if you can remember.

Did you come up with any? These are the ones that we really focused on recently.

Double number lines.

Here is an example of a double number line where we're using it to convert between percentages and scores.

Ratio tables.

Ratio tables are a really useful way of setting out our work to show if something has a multiplicative relationship.

Example here, we've got the mass and the percentage, so working out what the percentages are in comparison to the mass.

And graphs.

We can represent a multiplicative relationship using a graph.

So for example, here we've got the number of boxes plotted with the cost of buying that number of boxes.

Does that graph show a multiplicative relationship? It does.

And remember, a good way to check whether something has a multiplicative relationship is to check if we double one item does the other double.

From this graph, we can see if I buy one box, it costs two pounds.

Let's double the number of boxes to two boxes and we can see the cost is four pounds, so we've doubled the boxes and the cost is doubled.

Therefore, this shows a multiplicative relationship.

Here is a graph showing the relationship between two currencies used in a computer game, so we have zigs and we have zags.

This is a graphical representation of the relationship between the two.

But does it show a multiplicative relationship between zigs and zags? Let's check.

Two zigs is equal to three zags.

Two zigs is three zags.

Now let's double.

Double the number of zigs to four and read off the number of zags, which is six.

We can see that yes, there is a multiplicative relationship between them.

I've doubled the number of zigs and that doubles the number of zags.

So yes, the zigs and zags are directly proportional to each other.

We could also represent this on a double number line and you'll be familiar with double number lines.

Here, we can see the zags on the top and the zigs on the bottom, and we can read off values from our double number line.

The quickest and easiest way, like I said, to decide if two variables are indirect proportion is by checking as one variable doubles, so does the other.

Does this table show a directly proportional relationship between A and B? We've got column for A, and a column for B.

Does it show a directly proportional relationship? Remember, directly proportional relationship, if it has a multiplicative relationship and that's constant throughout, then they are directly proportional to each other.

Let's take a look.

Two.

If I double two, I get four.

Now, check B.

If I double six, I get 12.

Remember I said, it must be a constant multiplicative relationship, so we do need to check another value.

I've chosen to check with the two and the 12.

What is my multiplicative relationship between two and 12? And that's multiplied by six.

Let's check on the right hand side for B, then, six multiplied by six is 36.

Yes, A and B do have a directly proportional relationship.

Does this table show a directly proportional relationship between A and B? Here's our table.

We've got A, 1, 2, 3, 4, and B, 3, 4, 5, 6.

Do you think that shows a directly proportional relationship? Let's check.

What's my multiplicative relationship between one and two? That's multiplied by two.

What's my multiplicative relationship between three and four? And that's multiplied by four over three.

We can see here the multiplicative relationship is not the same, so, therefore, these are not directly proportional to each other.

They do not have a directly proportional relationship.

Like I said, the multiplier between values of A is not the same as the multiplier between the values of B.

Your turn now.

True or false? This table shows a directly proportional relationship between A and B.

You've got your table of values.

I'd like you to decide is the answer true or false? And not only that, you then need to decide what your justification is.

Why have you chosen true or false? Pause the video, and then when you've got your answer, come back.

What did you decide? This is false, and the correct justification is B, when A doubles B does not double.

If we have a look, we double two, if we look at A first, we double two to get four, but if I double seven, I don't get 13.

Therefore, it is not showing a directly proportional relationship between A and B.

Well done if you've got that right, but I'm sure you did.

Now you can have a go at some of these independently.

I've given you variables which are A and B, and I've represented these in a table just like we had in the previous examples.

Your job is to decide whether you think there is a directly proportional relationship between A and B.

A directly proportional relationship is one where the multiplicative relationship is constant throughout.

Pause the video, and when you've decided, remember no guessing, make sure that you can convince me of your answers and I'll be here waiting when you get back.

Okay, question number two.

Below are some variables which are directly proportional to each other.

A, Jun is painting his neighbor's fence.

He paints one fence every 40 minutes.

One fence panel.

Andeep is making necklaces for the school fete.

He makes two necklaces every seven minutes.

C, five miles is equivalent to eight kilometres.

D, a box of tiles covering 0.

75 metres squared, costs 15 pounds, and E, Sophia's dog eats three tins of food every four days.

For each of those situations, I'd like you to draw a double number line and mark on at least three equivalents, and a ratio table.

So when you are drawing your equivalents and your ratio table, you'll be able to choose which values you are using, but remember the directly proportional relationship is given to you in each part of the question.

Pause the video and when you are ready, come back, and we'll check those answers for you.

Great work.

Now we can check our answers.

Firstly, let's start with question number one.

We were looking for tables that showed there was a directly proportional relationship between A and B.

A does show a directly proportional relationship.

A.

When A is three and we double that to six, B is 12, that doubles to 24.

Also, you could be looking for the multiplicative relationship between A and B rather than A and A, and B and B, and we can see here that all the way through the table to get from A to B, the multiplicative relationship is four.

Remember to check all values because it needs to be a constant multiplicative relationship.

What about B? B does not show a directly proportional relationship.

C does.

And I'm going to go horizontally.

My multiplier from A to B is seven D does.

This time my multiplier from A to B is 0.

5.

E doesn't.

And F does.

And here my multiplicative relationship between A and B is 0.

25.

How did you get on? Great.

And here are some example answers to the question 2.

I've got the fence and I've got the time.

The one that comes from the question is one fence every 40 minutes, so that's the first one, and then I have some equivalents.

So two hours would be, sorry, two fences would be 80 minutes, five, 200.

You probably got different values there.

And there's an example of something you may have in your ratio table.

B.

Here we can see necklaces and time.

From the question, we've got two necklaces every seven minutes and then I've just chosen some values to add in on my double number line.

And there is an example of a ratio table.

You should have the two and the seven, but you may have something other than a 10 and the 35.

C.

We've got miles and kilometres.

Again, those are some examples of values you may have on your double number line and your ratio table.

D.

We've got the area and the cost this time.

E, Sophia's dog.

So we've got the number of tins of dog food and the number of days.

Great.

Now we can move on to the last learning cycle, which is graphing of proportional relationships.

Now we have already started to look at that in the second learning, sorry, in the first learning cycle, but what we're going to do is look at that in more depth now.

The cost of strawberries is directly proportional to the number of boxes.

Three boxes of strawberries cost eight pounds, 25.

We can represent this situation on a graph.

I've decided to put my boxes horizontally and the cost vertically.

Here is a ratio table, so three boxes cost eight pounds, 25, we are told that in the question.

I can then choose any value I like as my other value in my ratio table.

I often choose just a double because double is nice and easy.

Let's look for the multiplicative relationship between the three and the six because that's going to be easier than the three and the eight pounds, 25.

And I can see my multiplicative relationship because I'd already said I've just doubled it.

It is multiplied by two, so the cost is going to multiply by two.

Because we know this because it is directly proportional, so, therefore, a constant multiplicative relationship.

And that gives me 16 pounds 50.

I can then plot these points onto my graph, but I'm gonna start with zero, zero.

If I buy zero boxes of strawberries, I'm going to be paying zero pounds.

Three boxes of strawberries cost me eight pounds 25, so I'm going to mark that onto my graph.

And six boxes are going to cost me 16 pounds 50.

So we go along to six and up to 16 pounds 50 and we mark that.

All of those crosses should be in a straight line because this shows a directly proportional relationship, so as one thing increases, the other increases at the same rate.

And then we can draw the line, joining those.

Notice I've made my line go across my entire graph.

I didn't just stop at the six.

We're able now, to answer this question, looking at the graph.

Can you spend exactly 20 pounds on strawberries? We'll give you a moment to think about that.

Can you spend exactly 20 pounds on strawberries? And also I'd like you to justify your answer.

What did you decide, yes or no? And the answer is, if we look, 20 pounds, this is obviously the cost, 20 pounds, let's draw a line making sure that it's perpendicular to the cost axis, across to our exchange line or our directly proportional line, and we read off.

And we can see that no, reading from the graph, the number of boxes is not an integer.

In order to be able to spend exactly 20 pounds that would need to land on an integer, we cannot buy seven and a bit boxes of strawberries.

So although the graph sometimes might show that there are values in between, based on the context of the question, you might not be able to just read off any values.

Your turn now then.

This graph shows the directly proportional relationship between milkshakes and cost.

Looks like a nice mint milkshake to me.

Can you spend exactly seven pounds on milkshakes? Pause the video.

Remember, I don't just want yes or no.

I want that justification as well to show me that you truly really understand how we can use this graph to answer that question.

So pause the video now, and come up when you're ready.

What did you decide? Let's take a look.

Seven pounds.

So we draw across and we go down.

So the answer was no.

Seven pounds is between the cost of two milkshakes and three milkshakes.

We haven't got quite enough to buy three milkshakes.

So the answer to that is, no.

If it doesn't land on an integer.

In the context of this question, we would have to say no.

Now we have some watermelons.

We know that variables which are directly proportional can be represented on a double number line and a graph.

The cost of watermelons is directly proportional to the number of watermelons bought.

That's true, isn't it? Because each watermelon has a constant price.

What does that look like on a double number line? It looks like this.

One watermelon costs two pounds, so two costs four pounds, and three costs six pounds.

If I rotate my top part of my double number line, you can see that actually this creates our graph, and we can draw our line on our graph.

You can see that the two things are very clearly linked.

I'd like you to match each graph to the situations they represent.

So we've got graphs, A, B, and C, and our situations are one box costs four pounds, one box costs 50 pence, and one box costs two pounds 50.

I'm not sure what we're buying, but it doesn't matter, we can answer this question without knowing that.

Pause the video, and then come back when you're ready.

How did you get on? Great work.

One box costs four pounds is B.

Hopefully you spotted that because if we've got across the bottom is the amount, find one read up, it's four.

One box costs 50 pence was C.

And again, if we re-go along to one box, and up, we can see that that's halfway between zero and one, which is 50 pence.

And then finally, obviously it matches up.

One box costs two pounds, 50, matches with A.

But remember, we should always double check.

So go along one, and it is halfway between two pounds and three pounds, which is two pounds 50.

Now the task.

You need to match each table to the correct graph, so very similar to what you've just done in that check for understanding, but this time I've given you the values in a graph.

And then I'd like you to use the graph or otherwise because you know how to use a ratio table, to fill in the missing values.

Pause the video, and then when you come back we will check those answers.

Fabulous.

Now let's have a look at question number two.

So this time, I'm gonna need you to draw or plot the graph of each of the following.

So you've got your table of values, you need to fill in the missing table, or sorry, you need to fill in the missing values using the fact we know they're directly proportional.

So that must mean they have a constant multiplicative relationship.

Remember, you can move vertically or horizontally.

Once you've got your points, you should then have four points that you can plot onto your graph and then join those points together, remembering to use a ruler and a pencil.

Now there's one on each slide, so I'll pause, let you have a go, and then when you come back we'll move on to the next question.

Pause the video now, and have a go at A.

Well done.

And now B.

And C.

And question number 3.

This time you are going to match each double number line to the correct graph.

So in the first question, you were matching the graph to the table, now you are matching the graph to the double number line.

Pause the video, and then come back.

I'm pretty certain this is the last question now, so stick with me just for a bit longer.

Super work.

Here are our answers.

We needed to match the table to the correct graph.

So the first table matched the graph C.

The second table matched the graph A.

The third table matched graph D.

And the fourth table matched to B.

And then the missing values in the first table were 7.

5 and 20.

In the second one it was nine and 12.

The third one was two and four, and then in the final table the missing values were 10 and 25.

Now if I went too quick for you, remember, you can pause the video, check those and come back when you're ready.

Question number two.

These are the values in your table.

So it should read three with nine, four with 12 and six with 18.

And then you can plot those points and I hope you've used your ruler and pencil to join those points together carefully.

B, your missing values work four, six, and eight.

And your graph should look like this, once you've plotted it.

C, the missing values were three, six, and 12, and that's what your graph should look like.

And then finally 3, we were matching the double number line to the correct graph.

The first double number line matched with graph D, the second one matched with C, the third one matched with B, and the final double number nine matched with graph A.

How did you get on? Great work.

Now, we'll summarise the learning from today's lesson.

If two variables have a constant multiplicative relationship, they are in direct proportion.

Remember, you've looked at multiplicative relationships lots and lots of times previously, so all we are really doing is saying that we have a special way of referring to those and that is that they are directly proportional.

Some examples of things that are directly proportional, converting between units of measure, recipes, so scaling those recipes, converting between units of currency, between dollars and pounds, buying different amounts of an item.

Variables which are directly proportional can be represented on a double number line, we looked at that, in a ratio table, and on a graph.

And we already knew that because we knew these were ways of showing variables which have a multiplicative relationship, and we now know we're going to refer to those things as directly proportional.

Great work, today's lesson.

Well done.

I'm really glad that you decided to join me and I look forward to seeing you again really soon.

Goodbye.